Individualized physiology-based digital twin model for sports performance prediction: a reinterpretation of the Margaria–Morton model

Performance in many racing sports depends on the ability of the athletes to produce and maintain the highest possible work i.e., the highest power for the duration of the race. To model this energy production in an individualized way, an adaptation and a reinterpretation (including a physiological meaning of parameters) of the three-component Margaria–Morton model were performed. The model is applied to the muscles involved in a given task. The introduction of physiological meanings was possible thanks to the measurement of physiological characteristics for a given athlete. A method for creating a digital twin was therefore proposed and applied for national-level cyclists. The twins thus created were validated by comparison with field performance, experimental observations, and literature data. Simulations of record times and 3-minute all-out tests were consistent with experimental data. Considering the literature, the model provided good estimates of the time course of muscle metabolite concentrations (e.g., lactate and phosphocreatine). It also simulated the behavior of oxygen kinetics at exercise onset and during recovery. This methodology has a wide range of applications, including prediction and optimization of the performance of individually modeled athletes.

Here, we detail some of the calculations for the modified M-M model.These calculations are an adaptation of those presented in the description of Morton's original model 1-3 , but extended to the modified M-M model we introduced in our article.We will use the same notations as described in the paper.The two governing equations of the system are: Where V stands for volume, V for volume variation with respect to time, and D for flow rate. 1 is the indicator function.
From Equation 1we can describe the behavior of the system in the three work-rate regions at constant power.We will express the kinetics of l (lactate accumulation) and V O 2 , as well as their values and associated P physio at the thresholds between the regions.

Moderate work rate -Below LT1
We have h < θ and Thus the surface of the G compartment remains in the capillary region.There is a linear relation between V O 2 and D O→P : V O 2 = D O→P /C 1 .Therefore: Similarly with the classical Morton model, one can get a second-order differential equation on either l or h.
l has the following form: h has the following form: In mmol/kg w. w. .
Therefore we can predict the V O 2 evolution: In mL O 2 /s.

Upper limit of the work rate region -LT1
Recalling equation Equation 1with dl/dt = 0 and at equilibrium (dh/dt = 0) we have: The largest possible value in this first case is reached when h −→ θ .Therefore: Heavy work rate -Between LT1 and P crit Initially, h < θ and we will therefore observe the same behaviors as described previously until h = θ at a given time t 1 .Then we have the following governing equations: We can get a second-order differential equation on l or h.The one on l has the same form as the one for low work rate but A T is replaced by A G .
With initial conditions l(t 1 ) = θ and dl/dt(t 1 ) = M G A G h−θ 1−λ we can similarly analytically solve the second-order differential equations.

V O 2 Kinetics
From h we can express the V O 2 evolution: In mL O 2 /s.
Upper limit of the work rate region -P meca crit P meca crit corresponds to the limit case in which we can still observe an equilibrium, ie.we do not reach exhaustion and satisfy the constrain.Recalling Equation 1 at equilibrium (dh/dt = 0 and dl/dt = 0, h = l > θ ).The largest possible value in this case is reached when P physio −→ P max .